arXiv:2404.19183v1 [math.AG] 30 Apr 2024
Logarithmic geometry and Frobenius
Kazuya Kato, Chikara Nakayama, Sampei Usui
May 1, 2024
Abstract
Based on the logarithmic algebraic geometry and the theory of Deligne systems,
we define an abelian category of ℓ-adic sheaves with weight filtrations on a logarithmic scheme over a finite field, which is similar to the category of variations
of mixed Hodge structure. We consider asymptotic behaviors and simple cases of
higher direct images of objects of this category. This category is closely related to
the monodromy-weight conjecture.
Introduction
In this paper, based on the theory of logarithmic étale topology ([14], [15]) and the theory
of Deligne systems ([17], [1]), we define an abelian category AX of ℓ-adic sheaves with
weight filtrations on an fs logarithmic scheme X over a finite field.
This category is similar to the category of variations of mixed Hodge structure. We
describe asymptotic behaviors of objects of AX (Theorem 5.3), which are similar to asymptotic behaviors of variations of mixed Hodge structure in degeneration. We consider simple
cases (Propositions 6.5, 6.11) of higher direct images of objects of AX .
This category is closely related to the monodromy-weight conjecture (1.4, 1.6).
In Section 1, we explain our main ideas, main definitions, and results of this paper.
In Section 2, we consider admissible actions of cones and review the theory of Deligne
systems. In Section 3, we review the space of ratios used in Sections 4 and 5. In Section 4,
we discuss basic things about the category AX . In Section 5, we state and prove Theorem
5.3. In Section 6, we consider higher direct images of objects of AX and study some
examples.
K. Kato was partially supported by NFS grants DMS 1303421, DMS 1601861, and
DMS 2001182. C. Nakayama was partially supported by JSPS Grants-in-Aid for Scientific
Research (B) 23340008, (C) 16K05093, and (C) 21K03199. S. Usui was partially supported
by JSPS Grants-in-Aid for Scientific Research (B) 23340008, (C) 17K05200, and (C)
22K03247.
MSC2020: Primary 14A21; Secondary 14F06, 14G15
Keywords: logarithmic geometry, Deligne systems, asymptotic behaviors, monodromy-weight conjecture
1
Logarithmic geometry and Frobenius
1
Logarithmic local systems and Frobenius
In this Section 1, we introduce our main ideas and main results.
Let k be a finite field.
Let ℓ be a prime number which is not the characteristic of k. We fix an isomorphism
of fields ι : Qℓ ∼
= C.
Let X be an fs log scheme of finite type over k.
1.1. Let BX be the category of smooth Qℓ -sheaves H on X for the log étale topology
endowed with a smooth finite increasing filtration (which we denote by W ). Let AX be
the full subcategory of BX consisting of objects which satisfy the following conditions 1
and 2 for every closed point x of X.
1. The local monodromy of H at x is unipotent and is admissible with respect to W .
Here we use the fixed isomorphism ι : Qℓ ∼
= C to define the admissibility (2.3) of the
action of R≥0 -cones.
2. For the monodromy cone σ(x) at x, for the relative monodromy filtration W (σ(x))
W (σ(x))
H is of Frobenius weight w as a representation of Gal(k/k).
and for every w ∈ Z, grw
Here we use the fixed isomorphism ι to define the weight of the Frobenius. (For the
monodromy cone σ(x), see 1.8. For W (σ(x)), see 2.3.)
Remark 1.2. (1) Not only the Frobenius weight in the condition 2, the admissibility in
the condition 1 depends on the choice of the isomorphism ι (see 4.7). If X is of log rank
≤ 1, this admissibility is independent of the choice.
(2) There are two kinds of log étale topologies: the kummer log étale topology considered in [14] and the full log étale topology considered in [15]. We can use any one of
them, that is, the theory in this paper becomes the same for both (BX are the same, AX
are the same, and so on) by [15] Theorem 5.17. But to fix ideas, we use the kummer log
étale topology below.
1.3. We hope that AX is an analogue of the category of variations of mixed Hodge
structure and has similar properties to the latter.
For example, the category AX is an abelian category. This follows from the theory of
Deligne ([1] Lemma 6.33) on Deligne systems, as is explained in 4.5.
We have a theorem (Theorem 5.3) on the asymptotic behavior of an object of AX ,
which is similar to the asymptotic behavior of a variation of mixed Hodge structure in
degeneration. This is obtained from the theory of Deligne systems by using the space of
ratios in Section 3. Theorem 5.3 is useful in the study of the asymptotic behaviors of
the regulator maps and height pairings in degeneration. This study is illustrated in [8]
Remark 2.4.18 and will be explained elsewhere.
The category AX is related to the monodromy-weight conjecture as in the following
Conjecture 1.4 and 1.6.
Conjecture 1.4. Let f : X → Y be a projective log smooth vertical saturated morphism
of fs log schemes over k. Then Rm f∗ (Qℓ ) is an object of AY of pure W -weight m.
1.5. By proper base change theorem ([14] Theorem (5.1), [15] Theorem 6.1), Conjecture
1.4 is reduced to the case where Y is an fs log point.
2
Logarithmic geometry and Frobenius
1.6. In the case where Y is the standard log point, Conjecture 1.4 implies the monodromy
weight conjecture. In fact, if K is a non-archimedean local field with residue field k and if X
is a projective scheme over OK with semi-stable reduction, then Rm f∗ Qℓ for X = X ⊗OK k
and Y = Spec(k) with the canonical log structures is identified with the representation
m
Hét
(X ⊗OK K, Qℓ ) of Gal(K/K), and the monodromy-weight conjecture for X is exactly
that Rm f∗ Qℓ is an object of AY and is pure of W -weight m.
Remark 1.7. (1) Conjecture 1.4 becomes not true if we replace the assumption of the
projectivity by the properness. In fact, for the special fiber X → Y of a formal model
of the Hopf surface over a non-archimedean local field (X is algebraic though the Hopf
surface is not algebraic but only analytic), R1 f∗ Qℓ is Qℓ of weight 0, not of weight 1.
(2) Without the condition saturated, we may not have the unipotence of local monodromy.
1.8. We give some pictures of the categories BX and AX in the case where x is a point.
Let X be an fs log point whose underlying scheme is Spec(k) (k is a finite field). We
denote X also by the small letter x because it is a point. Let k be a separable closure of
x and let x = Spec(k). We have an exact sequence
1 −−−→
−−−→ Gal(x(log)/x) −−−→ Gal(x/x) −−−→ 1.
Gal(x(log)/x)
×
Hom ((MXgp /OX
)x , Z) ⊗ Ẑ(1)′
π1log (x)
Gal(k/k)
Here x(log) is Spec(k) endowed with the log structure obtained by adding n-th roots of
the log structure of x for all n ≥ 1 which are not divisible by the characteristic of k (i.e.,
x(log) is the associated log separably closed field in the sense of [14] Definition (2.5)), and
Ẑ(1)′ is the product of Zℓ′ (1), where ℓ′ ranges over all prime numbers which are different
from the characteristic of k. If q denotes the order of the finite field k and if F is an
element of π1log (x) whose image in Gal(k/k) is the q-th power map, we have F γF −1 = γ q
for all γ ∈ Gal(x(log)/x).
By taking the stalk at x(log), an object of BX is identified with a finite dimensional
representation of π1log (x) over Qℓ endowed with a π1log (x)-stable finite increasing filtration
W . The action of Gal(x(log)/x) on this stalk is called the local monodromy at x.
×
add
The cone σ(x) := Hom ((MX /OX
)x , Radd
≥0 ), where R≥0 = {x ∈ R | x ≥ 0} with the
additive structure, is called the monodromy cone at x. The admissibility of the local
monodromy in 1.1 means the admissibility of the logarithm action of this monodromy
cone.
1.9. Assume that X = {x} is the standard log point Spec(k) whose log structure is
associated to N → k ; n 7→ 0n .
Then Gal(x(log)/x) ∼
= Gal(Ktame /K) for a non-archimedean local field K with the
residue field k and for its maximal tame extension Ktame .
×
Let N ∈ σ(x) be the standard generator of Hom ((MX /OX
)x , N) = Hom (N, N) = N.
For r ≥ 0, we define an object Sr of BX . Fix a splitting of the surjective homomorphism
log
π1 (x) → Gal(k/k). Let
(1)
S1 = Qℓ ⊕ Qℓ (−1)
Logarithmic geometry and Frobenius
as a representation of Gal(k/k). Let e1 = (1, 0) ∈ S1 and let e2 = (0, e), where e is any
fixed base of Qℓ (−1). Define the action of π1log (x) on S1 by N(e2 ) = e1 and N(e1 ) = 0.
Define W of S1 to be pure of weight 1. Then S1 is an object of AX . Let Sr be the r-th
symmetric power of S1 . This is an object of AX of pure W -weight r.
As Proposition 1.10 below shows, Sr are simple objects of AX . This is remarkable
because we have for example an exact sequence
0 → Qℓ → S1 → Qℓ (−1) → 0
of ℓ-adic sheaves on the log étale site of X which tells that S1 is not simple as an ℓ-adic
sheaf.
For w ∈ Z, let Aw be the full subcategory of AX consisting of all objects which have
pure W -weight w.
On the other hand, for a smooth Qℓ -sheaf H on X with no log structure, if H has
pure Frobenius weight w, we regard H as an object of AX with pure W -weight w in the
natural way. Let A′w be the category of families (Hr )r≥0 of smooth Qℓ -sheaves Hr on X
with no log structure such that Hr has pure Frobenius weight w − r and such that Hr = 0
for almost all r. We have a functor
M
(2)
Φ : A′w → Aw ; (Hr )r 7→
Sr ⊗ H r .
r
Proposition 1.10. Assume that X is the standard log point.
(1) The functor Φ gives an equivalence of categories
≃
A′w → Aw .
(2) Let (Hr )r be an object of A′w and let H = Φ((Hr )r ). Then H is semi-simple in
AX if and only if Hr are semi-simple for all w. H is simple in AX if and only if there is
an r such that Hr is of rank 1 and such that Hs = 0 for all s 6= r.
(3) A W -pure object H of AX is semi-simple if and only if H is semi-simple as a
representation of Gal(k/k) (forgetting the local monodromy).
This will be proved in 4.6.
×
1.11. We are using the R≥0 -cone σ(x) = Hom ((MX /OX
)x , Radd
≥0 ). If we use the Q≥0 -cone
×
add
σQ (x) := Hom ((MX /OX )x , Q≥0 ) instead of σ(x), we have a similar but different theory.
Concerning this, see 4.8.
1.12. The category AX has a crystalline analogue and a p-adic Hodge analogue. We will
consider them in a sequel of this paper.
Remark 1.13. One of the authors (K. Kato) published a paper [7] concerning Deligne
systems. It is shown in [1] Section 6 that this paper [7] is wrong. He hopes that [7] is
never used by any person. He hoped that the paper [7] could be used in the study of
degeneration of a family of motives over a non-archimedean local field ([7] 1.9). We hope
that Theorem 5.3 in our present paper is on the right way for that purpose.
Logarithmic geometry and Frobenius
2
Reviews on admissible actions of cones and on Deligne
systems
We review relative monodromy filtration, admissibility of an action of a cone, Deligne
splitting, and Deligne systems (See [17]).
2.1. Let V be a vector space endowed with a finite increasing filtration W , and let
N : V → V be a nilpotent linear map such that NWw ⊂ Ww for any w ∈ Z. A finite
increasing filtration W on V is called the relative monodromy filtration of N with respect
to W if it satisfies the following two conditions (i) and (ii).
(i) NWw ⊂ Ww−2 for any w ∈ Z.
∼
W
W
W =
(ii) For every w ∈ Z and every integer r ≥ 0, the map N r : grW
w+r grw → grw−r grw is
an isomorphism.
The relative monodromy filtration W of N with respect to W is unique if it exists ([3]
1.6.13).
2.2. By a finitely generated cone, we mean a subset σ of a finite dimensional R-vector
space V such that σ = R≥0 N1 + · · · + R≥0 Nn for some N1 , . . . , Nn ∈ V . Here R≥0 = {x ∈
R | x ≥ 0}.
We denote the R-linear span of σ in V by σR .
2.3. Let σ be a finitely generated sharp cone and let V be a finite dimensional C-vector
space endowed with a finite increasing filtration W . By an admissible action of σ on V ,
we mean an R-linear map h : σR → EndC (V ) satisfying the following conditions (i)–(iii).
(i) h(N)h(N ′ ) = h(N ′ )h(N) for all N, N ′ ∈ σR and N(Ww ) ⊂ Ww for all N ∈ σR and
w ∈ Z.
(ii) For every N ∈ σR , h(N) is nilpotent.
(iii) There is a family (W (τ ))τ of finite increasing filtrations W (τ ) on V , where τ
ranges over all faces of σ, satisfying the following conditions (iii-1), (iii-2), (iii-3).
(iii-1) For every face τ of σ and for every N ∈ σR and w ∈ Z, we have N(W (τ )w ) ⊂
W (τ )w .
(iii-2) W ({0}) = W .
(iii-3) For any faces τ , τ ′ , τ ′′ of σ such that τ ⊃ τ ′ and for any w ∈ Z and any element
N of the interior of τ , the restriction of W (τ ) to W (τ ′′ )w is the relative monodromy
filtration of the restriction of N to W (τ ′′ )w with respect to the restriction of W (τ ′ ) to
W (τ ′′ )w .
Remark 2.4. Concerning the above condition (iii):
(1) By the conditions (iii-2) and (iii-3), for any face τ of σ and for any element N
of the interior of τ , W (τ ) is the relative monodromy filtration of N with respect to W .
Hence the family (W (τ ))τ is unique if it exists.
(2) (1) tells that for any face τ of σ and for any N ∈ τR and w ∈ Z, we have
N(W (τ )w ) ⊂ W (τ )w−2. This is because the interior of τ generates τR as an R-linear
space.
2.5. For an admissible action σ on V and for a face τ of σ, we have NW (τ )w ⊂ W (τ )w
for all w ∈ Z and for all N ∈ σR .
Logarithmic geometry and Frobenius
In fact, for an element N ′ of the interior of τ , W (τ ) is the relative monodromy filtration
of N ′ with respect to W , and hence exp(N)W (τ ) is the relative monodromy filtration of
exp(N)N ′ exp(−N) = N ′ with respect to exp(N)W = W , and hence coincides with W (τ ).
2.6. The admissibility is stable under taking direct sums, tensor products, and duals.
2.7. We review the theory of Deligne splitting ([17] Theorem 1) due to Deligne.
We return to the setting in 2.1. Assume that the relative monodromy filtration W of
N with respect to W exists, and assume that we are given a splitting Y of W satisfying
the following two conditions.
(i) N is of weight −2 for Y .
(ii) Y is compatible with W . That is, the action of Gm on V associated to Y (a ∈ Gm
acts on the part of V of weight w for Y by aw ) keeps W .
Then there is a unique splitting Y ′ of W , which we call the Deligne splitting, satisfying
the following two conditions.
(i) Y ′ is compatible with Y . That is, there is an action of G2m on V in which the first
Gm gives Y ′ and the second Gm gives Y .
(ii) Let
M
W
End(V ) ∼
grW
=
m grw End(V )
w,m
L
W
be the isomorphism given by (Y ′ , Y ), and write the image of N ∈ End(V ) in w,m grW
m grw End(V )
P
W
W
as
d≥0 Nd with Nd ∈ gr−2 gr−d End(V ). Then for d ≥ 1, Nd belongs to the primiW
(d−1)/2
W
tive part of grW
: grW
−2 gr−d End(V ), that is, Nd is killed by Ad(N)
−2 gr−d End(V ) →
W
grW
−2d gr−d End(V ).
2.8. In 2.8 and 2.9, we review the theory of Deligne systems ([17] Section 2, [1] Section
6).
An n-variable Deligne system is a system (V, (W j )0≤j≤n , (Nj )1≤j≤n , Y ), where V is a
finite dimensional vector space, W j (0 ≤ j ≤ n) are finite increasing filtrations on V ,
Nj : V → V (1 ≤ j ≤ n) are mutually commuting nilpotent linear operators, and Y is a
splitting of W n satisfying the following conditions (i)–(iii).
(i) For 1 ≤ j ≤ n, 0 ≤ i ≤ n and w ∈ Z, the restriction of W j to Wwi is the relative
monodromy filtration of the restriction of Nj to Wwi with respect to the restriction of
W j−1 to Wwi .
j
(ii) Ni ∈ W0j End(V ) for all i and j, and Ni ∈ W−2
End(V ) if i ≤ j.
(iii) Nj are of weight −2 for Y for all 1 ≤ j ≤ n, and Y is compatible with W j for all
0 ≤ j ≤ n.
The category of n-variable Deligne systems is an abelian category ([1] Lemma 6.33,
due to Deligne).
2.9. ([17] Theorem 2, due to Deligne.)
Let (V, (W j )0≤j≤n , (Nj )1≤j≤n , Y ) be an n-variable Deligne system. Then there is a
unique action ρ of Gm × SL(2)n on V characterized by the properties (i) and (ii) below.
For 0 ≤ j ≤ n, define the splitting Y j of W j by downward induction on j in the
following way. Y n := Y . For 1 ≤ j < n, since W j+1 is the relative monodromy filtration
Logarithmic geometry and Frobenius
of N j+1 associated to W j , the splitting Y j+1 of W j+1 (given by downward induction on
j) and Nj+1 define a splitting of W j by the theory of Deligne splitting (2.7). For each
1 ≤ j ≤ n, let N̂j be the component of Nj of degree 0 for the splittings Y i of W i for
1 ≤ i < j.
0 1
(i) The Lie action Lie(ρ) sends the matrix
in the j-th sl(2) in sl(2)n to N̂j .
0 0
(ii) For 0 ≤ j ≤ n, let τj be the action of Gm on V given by the splitting Y j of W j .
Then Gm in Gm × SL(2)n acts via τ0 , and for 1 ≤ j ≤ n, τj (a) = τ0 (a)ρ(b) for a ∈ Gm .
Here b is the element of SL(2)n whose i-th component is diag (a−1 , a) if 1 ≤ i ≤ j and is
1 if j < i ≤ n.
2.10. Let σ and V be as in 2.3 and assume that we are given an admissible action of σ
on V . Assume that we are given a sequence of faces σ0 ( σ1 ( · · · ( σn = σ, an element
Nj of the interior of σj for each 1 ≤ j ≤ n, and a splitting Y of W (σ) such that Nj are
of weight −2 for Y for 1 ≤ j ≤ n and Y is compatible with W (σj ) for 0 ≤ j ≤ n.
Then we have a Deligne system (V, (W j )0≤j≤n , (Nj )1≤j≤n , Y ), where W j := W (σj ),
and hence an action of Gm × SL(2)n on V by 2.9.
2.11. In 2.10, let τ be a face of σ. Then the action of Gm × SL(2)n on V keeps W (τ ).
In fact, for each w ∈ Z and for U := W (τ )w , (U, (W j |U ), (Nj |U ), Y |U ) is a Deligne
system by the condition (iii-3) in 2.3, and the associated action of Gm × SL(2)n on U is
compatible with the action of Gm × SL(2)n on V by the characterizations of these actions.
3
Reviews on the space of ratios
We review the space of ratios defined in [12] Section 4 and used in the study of degenerating
Hodge structures.
3.1. Definition ([12] 4.1.3). Let S be a sharp fs monoid. We denote the semi-group law
of S multiplicatively.
The space of ratios R(S) is the set of all maps (S × S) r {(1, 1)} → [0, ∞] satisfying
the following conditions (i)–(iii).
(i) r(f, g) = r(g, f )−1.
(ii) r(f, g)r(g, h) = r(f, h) if {r(f, g), r(g, h)} =
6 {0, ∞}.
(iii) r(f g, h) = r(f, h) + r(g, h).
We endow R(S) with the topology of simple convergence. It is compact.
3.2. Example. R(N2 ) is homeomorphic to the interval [0, ∞] as a topological space. In
fact, if (qj )j=1,2 denotes the standard base of N2 , r ∈ R(N2 ) corresponds to r(q1 , q2 ) ∈
[0, ∞].
3.3. There is a bijection between R(S) and the set of all equivalence classes of families
((σj )1≤j≤n , (Nj )1≤j≤n ), where n ≥ 0, σj are faces of the cone σ := Hom (S, Radd
≥0 ) such
that {0} := σ0 ( σ1 ( · · · ( σn = σ and Nj is an element of the interior of σj . Two such
families ((σj )j , (Nj )j ) and ((σj′ )j , (Nj′ )j ) are equivalent if and only if σj′ = σj for 1 ≤ j ≤ n
and there are cj ∈ R>0 such that Nj′ ≡ cj Nj mod σj−1,R for 1 ≤ j ≤ n. (Cf. [12] 4.1.6.)
Logarithmic geometry and Frobenius
In fact, for such a family ((σj )j , (Nj )j ), the corresponding r ∈ R(S) is as follows. Let
S = S (0) ) S (1) ) · · · ) S (n) = {1} be the sequence of faces of S for which σj is the
annihilator of S (j) . Then for (f, g) ∈ S × S r {(1, 1)}, if j is the smallest integer such
that {f, g} is not contained in S (j) , r(f, g) = Nj (f )/Nj (g).
3.4. Let R(S)1 be the subset of R(S) consisting of all elements such that the corresponding
sequences of faces of σ (3.3) are of length one, that is, n = 1.
If S = {1}, R(S)1 is empty.
Assume S =
6 {1}, and let σ ◦ be the interior of σ. Then, we have a bijection ψ :
◦
σ /R>0 → R(S)1 defined by ψ(N)(f, g) = N(f )/N(g) (N ∈ σ ◦ ), and ψ(N) corresponds
in 3.3 to the family (σ, N).
If S =
6 {1}, R(S)1 is a dense open subset of R(S). The density is provedPas follows.
Let r ∈ R(S) be the class of ((σj )1≤j≤n , (Nj )1≤j≤n ). Then r is the limit of ψ( nj=1 yj Nj ),
where yj ∈ R>0 and yj /yj+1 → ∞ for 1 ≤ j < n. The openness is shown as follows.
Assume that f1 , . . . , fm (fj ∈ S r {1}) generate S. Then r ∈ R(S) belongs to R(S)1 if
and only if r(fi , fj ) ∈ (0, ∞) for every i, j.
3.5. If x is an fs log point, we denote the space R((Mx /Ox× )x ) by x[:] and R((Mx /Ox× )x )1
by x[:],1 .
In the case where the underlying scheme of x is Spec(C), the space x[:] is important
in log Hodge theory to treat SL(2)-orbits. See [12] Theorem 4.5.2 and [13] Section 4.
3.6. Let x be an fs log point. To work on the topological space x[:] as we will do in
Section 5 may give the impression that we work in the category of topological spaces
leaving algebraic geometry. But this is not a correct feeling. If xval denotes the inverse
limit of log blowing ups of x, we have a surjective continuous map xval → x[:] such that the
topology of x[:] is the quotient topology of the topology of xval ([12] 4.1.11, 4.1.12). Hence
for an object H of Ax , the behavior of H on the topological space x[:] gives information of
the behavior of H on the space xval of algebraic nature. Since x and xval are identical for the
log étale topology (the full log étale topology ([15]), not the kummer log étale topology),
the behavior of H on the topological space x[:] gives information on the behavior of H on
the log étale site of x.
4
The category AX
We discuss basic facts about the category AX .
Lemma 4.1. Let x be an fs log scheme whose underlying scheme is Spec(k) for a finite
field k. Let H be an object of Ax and let V be the stalk of H.
Let τ be a face of σ(x), and let W (τ ) be the weight filtration on V associated to τ
(2.3). Then W (τ ) is invariant under the action of π1log (x).
This is a stronger version of 2.5 in the present situation.
Proof. Let q be the order of the finite field k, and let F ∈ π1log (x) be an element whose
image in Gal(k/k) is the q-th power map. Let N be an element of the interior of τ . Then
since W (τ ) is the relative monodromy filtration of N with respect to W , F W (τ ) is the
Logarithmic geometry and Frobenius
relative monodromy filtration of F NF −1 with respect to F W = W . Since F NF −1 = qN
(1.8), F W (τ ) coincides with W (τ ).
Since these F generate the group π1log (x), W (τ ) is invariant under the action of π1log (x).
Lemma 4.2. Let x, k, H and V be as in Lemma 4.1. Fix a splitting of π1log (x) →
Gal(k/k), and let Y be the splitting of W (σ(x)) by the Frobenius weights for the action
of Gal(k/k). Then the following hold.
(1) For every N ∈ σ(x)R , N : V → V is of weight −2 for Y .
(2) For every face τ of σ(x), Y is compatible with W (τ ).
Proof. Let F be the q-th power Frobenius in Gal(k/k) which acts on V via the splitting.
(1) follows from F NF −1 = qN.
We prove (2). For each c ∈ C× , if Vc denotes the generalized eigenspace of F in V for
the eigenvalue c, the projection V → Vc is given by a polynomial in F . L
By Lemma 4.1,
this projection sends W (τ )w into W (τ )w for each w, and hence W (τ )w = c W (τ )w ∩ Vc .
Hence Y is compatible with W (τ ).
4.3. Let the situation be as in Lemma 4.2.
Assume that we are given a sequence of faces σ0 ( σ1 ( · · · ( σn = σ and an element
Nj of the interior of σj for each 1 ≤ j ≤ n. Here σ0 need not be {0}.
Then by 2.10 and Lemma 4.2, we have a Deligne system (V, (W j )0≤j≤n , (Nj )1≤j≤n , Y ),
where W j := W (σj ), and hence an action of Gm × SL(2)n on V by 2.9. By 2.11, for every
face τ of σ(x) and for every w ∈ Z, W (τ )w is stable under this action of Gm × SL(2)n .
4.4. Let the situation be as in Lemma 4.2. We consider the space of ratios x[:] . Let
µ ∈ x[:] .
Recall that µ is the equivalence class of a family ((σj )1≤j≤n , (Nj )1≤j≤n ), where σj are
faces of σ(x) such that {0} = σ0 ( σ1 ( · · · ( σn = σ(x) and Nj is an element of the
interior of σj (see 3.3).
As in 4.3, this family ((σj )1≤j≤n , (Nj )1≤j≤n ) determines a representation of Gm ×
SL(2)n .
By the characterization of Deligne splitting reviewed in 2.7, for 1 ≤ j ≤ n, the splitting
Y j−1 of W j−1 in 2.9 depends only on Nj mod σj−1,R . Because elements of σj−1,R have only
weights ≤ −2 parts for Y j−1 , N̂j in 2.9 depends only on Nj mod σj−1,R . Hence this action
of Gm × SL(2)n is determined by ((σj )1≤j≤n , (Nj mod σj−1,R )1≤j≤n ).
If we replace Nj by aj Nj (1 ≤ j ≤ n) for aj ∈ R>0 , ρ is changed by its conjugate by
√
√
the action of the element of SL(2, C)n whose j-th component is diag( aj , 1/ aj ). Thus
µ determines the representation of Gm × SL(2)n modulo this conjugacy.
This is very similar to the relation between the space of ratios and SL(2)-orbits in
Hodge theory in [12] Theorem 6.3.1 (1), [13] Section 4.
4.5. Let X be an fs log scheme over a finite field.
We prove that AX is an abelian category. This is closely related to the fact that the
category of n-variable Deligne systems is an abelian category (2.8) and proved in a similar
way. We also prove the following (1) and (2).
Logarithmic geometry and Frobenius
(1) If H → H ′ is a morphism in AX , then at each closed point x ∈ X, for every face
σ of the monodromy cone σ(x) and w ∈ Z, the image of W (σ)w H in H ′ coincides with
the intersection of the image of H and W (σ)w H ′ .
(2) For any exact sequence 0 → H ′ → H → H ′′ → 0 in AX , at each closed point
x ∈ X, the sequence 0 → W (σ)w H ′ → W (σ)w H → W (σ)w H ′′ → 0 is exact for every w
and for every σ as in (1).
In the following proof, when we work at a closed point x of X, we will denote the
stalks of H, H ′ , etc., just by the same notation H, H ′ , etc.
We prove (1). Apply 4.3 to the case where n = 1, σ0 = σ, σ1 = σ(x), and N1 is an
element of the interior of σ(x). Then this defines actions of Gm on H and H ′ splitting
W (σ), and the homomorphism H → H ′ is compatible with these actions. This proves
(1).
We prove that AX is an abelian category.
First, for a morphism H → H ′ of AX , we prove that the kernel K with the induced
W and the cokernel C with the induced W are objects of AX . We consider the kernel.
For x ∈ X and for each face τ of σ(x), let W (τ )K be the filtration on K induced by the
filtration W (τ )H. We prove that we have an admissible action with the family (W (τ )K)τ
of relative monodromy filtrations. It is sufficient to prove that the condition (iii-3) in 2.3
is satisfied. Let τ, τ ′ , τ ′′ be as in this condition (iii-3).
Apply 4.3 to the case where n = 2, σ0 = τ ′ , σ1 = τ , σ2 = σ(x), N1 is an element of
the interior of τ , and N2 is an element of the interior of σ(x). Then we have actions of
SL(2)2 on H and H ′ , and the homomorphism H → H ′ is compatible with these actions.
This induces an action of SL(2)2 on K and on W (τ ′′ )w K for w ∈ Z (4.3). Hence for every
W (τ ) W (τ ′ )
W (τ ) W (τ ′ )
r, s ∈ Z with r ≥ 0, the map N1r = N̂1r : grs+r grs
W (τ ′′ )w K → grs−r grs
W (τ ′′ )w K
is an isomorphism.
The proof for the cokernel C is similar.
Next, we prove that the map from the image to the coimage is an isomorphism. But
this follows from (1) for W = W ({0}). This completes the proof of the statement that
AX is an abelian category.
We prove (2). Apply 4.3 to the case where σ0 = σ, σ1 = σ(x), and N1 is an element
of the interior of σ(x). Then we have the actions of Gm on H, H ′ , H ′′ splitting W (σ)
and we have the exact sequence of representations of Gm . Hence the induced sequence of
W (σ)w is exact.
4.6. We prove Proposition 1.10.
We prove (1). We write here the proof for the case w = 0. The method of the proof
for the general case is the same, but we assume this just to make the notation simple.
We fix a splitting of π1log (x) → Gal(k/k). Hence for every object H of AX L
, we have a
splitting of the Frobenius weight filtration. For an object H of AX , let H = w H [w] be
the decomposition, where H [w] is the part of H of Frobenius weight w (it is a sub Qℓ -sheaf
of H). We give the converse functor A0 → A′0 in two ways. First, define Ψ− : A0 → A′0 as
Ψ− (H) = (Hr )r≥0 , where Hr = Ker(N : H [−r] → H [−r−2] ). Next, let Ψ+ (H) = (Hr )r≥0 ,
where Hr = Ker(N r+1 : H [r] → H [−r−2] )(r) (this is the so-called primitive part). Then we
∼
∼
=
=
have a natural isomorphism Ψ+ → Ψ− given by N r : Ker(N r+1 : H [r] → H [−r−2] )(r) →
Ker(N : H [−r] → H [−r−2]). The composition Ψ− ◦ Φ is identified with the identity functor
Logarithmic geometry and Frobenius
of A′0 . We show that the composition Φ ◦ Ψ+ is isomorphic to the identity functor of
A0 . For an object H of AX , if (Hr )r≥0 = Ψ+ (H), we have the natural isomorphism
L
∼
=
−1 i j
i
r Sr ⊗ Hr → H given by Sr ⊗ Hr → H which sends j! e1 e2 ⊗ x (i + j = r) to N (x).
Here we identify Qℓ (i) with Qℓ by e⊗−i 7→ 1 for all i ∈ Z, where e is the base of Qℓ (−1)
which we fixed in 1.9.
(2) follows from (1).
(3) follows from (1) and (2).
4.7. We show that the admissibility in the condition 1 in 1.1 depends on the choice of
the isomorphism ι.
×
Let X be an fs log point (Spec(k), N2 ⊕ OX
). Define an object H of BX as follows.
As a representation of Gal(k/k), H = Qℓ ⊕ Qℓ (−1) with the base e1 , e2 as in 1.9. W of
H is pure of weight 1. Take non-zero elements aj (j = 1, 2) of Qℓ . Let the monodromy
operators Nj (j = 1, 2) be Nj (e1 ) = 0 and Nj (e2 ) = aj e1 . Then the local monodromy is
admissible if and only if x1 ι(a1 ) + x2 ι(a2 ) 6= 0 for every x1 , x2 ∈ R>0 . This condition is
equivalent to the condition that ι(a1 a−1
2 ) is not a negative real number, and hence depends
on the choice of ι (for example, it depends if a1 a−1
2 is a square root of 2).
In this situation, the local monodromy is admissible for every choice of ι if and only
if a1 a−1
2 is a totally positive algebraic number.
4.8. If we use the Q≥0 -cone σQ (x) (1.11) instead of the R≥0 -cone σ(x), and consider
the admissibility using only element of σQ (x), we do not need to use an isomorphism
Qℓ ∼
= C to define the admissibility. We still can prove that the version of AX for this
formulation is an abelian category (the proof is essentially identical with the proof given
above). However, this weaker admissibility does not give Theorem 5.3. For example, if x
is an fs log point with the log structure N2 as in 4.7, this weak admissibility√cannot give
the behavior of H around the point of x[:] ∼
= [0, ∞] (3.2) corresponding to 2 ∈ [0, ∞].
This is not nice for the applications explained in 5.6 below. This is the reason why we
like to use the R≥0 -cone.
5
Asymptotic behaviors
We prove a non-archimedean analogue Theorem 5.3 of the asymptotic behaviors of mixed
Hodge structures in degeneration. The latter is related to the SL(2)-orbit theorem in
Hodge theory. It has been studied by many people, for example, as in [2], [6], [10], [11],
[16], etc.
Theorem 5.3 is applied to non-archimedean geometry as in 5.6 below.
5.1. Let x be an fs log scheme whose underlying scheme is Spec(k) for a finite field k.
We assume that the log structure of x is not trivial.
Let H be an object of Ax and let V be the stalk of H.
We use the space x[:] of ratios and its dense open subset x[:],1 (3.5). Let µ ∈ x[:] . We
consider the behavior of H when ν ∈ x[:],1 converges to µ.
5.2. Let
E = End(V )
Logarithmic geometry and Frobenius
be the set of all linear operators on V regarded as a C-algebra.
Assume that µ ∈ x[:] is the class of ((σj )1≤j≤n , (Nj )1≤j≤n ) (3.3).
(0)
(1)
(n)
Let Mx = Mx ) Mx ) · · · ) Mx = Ox× = k × be the sequence of faces of Mx
(j−1)
which is dual to the filtration (σj )j on σ(x). Fix fj ∈ Mx (1 ≤ j ≤ n) which is in Mx
(j)
but not in Mx .
Replacing Nj (1 ≤ j ≤ n) by its R>0 -multiple, we assume that Nj : (Mxgp /Ox× )x → R
sends fj to 1 for 1 ≤ j ≤ n. Fix a splitting of π1log (x) → Gal(k/k). As in 4.4, we have a
splitting Y j of W j = W (σj ) for 0 ≤ j ≤ n and we have an action of Gm × SL(2)n on V .
Let N̂j ∈ E (1 ≤ j ≤ n) be as in 2.9. For 1 ≤ j ≤ n, let τj be the action of Gm on V
corresponding to Y j . For ν ∈ x[:],1, we define
N(ν) ∈ E,
t(ν) ∈ E ×
as follows.
Let Nν to be the unique element of the interior of σ(x) such that ν is the class of
(σ(x), Nν ) and such that Nν : (Mxgp /Ox× )x → R sends fn to 1. Let N(ν) be the image of
Nν in E.
Define
n−1
Y
t(ν) :=
τj (ν(fj+1 , fj )1/2 ) ∈ E × .
j=1
Theorem 5.3. When ν ∈ x[:],1 converges to µ ∈ x[:] , t(ν)−1 N(ν)t(ν) converges to
in E.
Pn
j=1 N̂j
This is an analogue of [11] Theorem 2.4.2 (ii) on SL(2)-orbits of mixed Hodge structures.
The following lemma is actually a part of the theory of Deligne systems, but we present
it here with proof because it plays a key role below.
Lemma 5.4. Let 1 ≤ j ≤ n and let N be an element of σj,R . Then N is purely of weight
−2 for Y i if j ≤ i ≤ n.
Proof. This is because N is a homomorphism of Deligne systems (V, (W i )j≤i≤n , (Ni )j<i≤n , Y n ) →
(V, (W i )j≤i≤n , (Ni )j<i≤n , Y n )(−1), where (−1) denotes the Tate twist, and the construction of the splittings Y i (j ≤ i ≤ n) is functorial.
5.5. We prove Theorem 5.3.
(j−1)
Let fj,λ ∈ Mx
(1 ≤ j ≤ n) be the elements such that for any j, (fj,λ)λ is a Q-basis
(j),gp
(j−1),gp
/Mx
). Let (Nj,λ)j,λ be the dual base of (fj,λ mod Ox× )j,λ in σ(x)R .
of Q ⊗ (Mx
Then
X
N(ν) =
ν(fj,λ , fn )Nj,λ.
j,λ
It is enough to prove that for each 1 ≤ j ≤ n, Ad(t(ν))−1
to N̂j in E.
We have P
(1) Nj ≡ λ µ(fj,λ, fj )Nj,λ mod σj−1,R .
P
λ
ν(fj,λ , h)Nj,λ converges
Logarithmic geometry and Frobenius
(0)
(<0)
(0)
Write Nj,λ = Nj,λ + Nj,λ , where Nj,λ is of τi -weight 0 for all i such that 1 ≤ i < j,
(<0)
and Nj,λ is of τi -weight ≤ 0 for all i such that 1 ≤ i < j and of τi -weight < 0 for some i
such that 1 ≤ i < j. By (1), we have
P
(0)
(2) N̂j = λ µ(fj,λ, fj )Nj,λ .
When ν converges to µ, ν(fj,λ , fj ) converges to µ(fj,λ, fj ).
Write t(ν) = t≥j (ν)t<j (ν) with
t≥j (ν) :=
n−1
Y
τi (ν(fi+1 , fi )
1/2
),
t<j (ν) :=
j−1
Y
τi (ν(fi+1 , fi )1/2 ).
i=1
i=j
By Lemma 5.4, τi (a)−1 Nj,λ = a2 Nj,λ if j ≤ i ≤ n. Hence
X
Ad(t≥j (ν))−1
ν(fj,λ , fn )Nj,λ
λ
=
n−1
Y
ν(fi+1 , fi )
i=j
=
X
X
ν(fj,λ , fn )Nj,λ =
λ
(0)
ν(fj,λ , fj )Nj,λ +
λ
X
ν(fj,λ , fj )Nj,λ
λ
X
(<0)
ν(fj,λ, fj )Nj,λ .
λ
(0)
(0)
(<0)
We have Ad(t<j (ν))−1 Nj,λ = Nj,λ , and Ad(t<j (ν))−1 Nj,λ tends to 0 when ν converges
P
to µ. Hence Ad(t(ν))−1 λ ν(fj,λ , fn )Nj,λ converges to N̂j .
5.6. Here, by using an example, we describe how Theorem 5.3 is applied to the nonarchimedean geometry.
Let K be a non-archimedean local field with finite residue field and let π be a prime
element of K.
We consider the following example. Endow Spec(OK [t]) with the log structure generated by π and t, and assume that our x is the closed point of Spec(OK [t]) at which π and
t have value 0. Let X be an open subscheme of Spec(OK [t]) containing x and let U be
the inverse image of X in Spec(K[t, t−1 ]). Assume that H comes from a smooth Qℓ -sheaf
H̃ on X. Then the restriction of H̃ to U is a smooth Qℓ -sheaf on the usual étale site.
×
If α ∈ K is near to 0 ∈ K, α defines a closed point Spec(K(α)) of U which we denote
by α. Let H̃(α) be the pullback of H̃ to α. We are interested in how the monodromy
operator of H̃(α) behaves when α tends to 0.
Let x(α) be the closed point of Spec(OK(α) ) with the standard log structure. If α is
near to 0, the ring homomorphism OK [t] → OK(α) ; t 7→ α induces a morphism x(α) → x
of log schemes. The monodromy cone σ(x(α)) is of rank one, and the homomorphism
σ(x(α)) → σ(x) sends a non-zero element to an element of the interior of σ(x). This
determines a point ν(α) of x[:],1 (3.4, 3.5).
In the homeomorphism x[:] ∼
= [0, ∞] ; r 7→ r(t, π) (3.2), ν(α) is identified with v(α) ∈
[0, ∞], where v is the valuation of K(α) normalized by v(π) = 1.
Since H̃(α) is identified with the pullback of H to x(α) as a representation of Gal(K(α)tame /K(α)) =
Gal(α(log)/α), the stalk of H̃(α) and the stalk of H are identified. We apply Theorem
5.3 by taking µ to be the point of x[:] corresponding to ∞ ∈ [0, ∞]. Then when α tends
Logarithmic geometry and Frobenius
to 0, ν(α) tends to µ. Take f1 = t, f2 = π in 5.2. Then N(ν(α)) in 5.2 is identified with
×
the action of the unique element Nα of σ(x(α)) which sends the class of π in Mx(α) /Ox(α)
to 1.
Hence Theorem 5.3 describes the behavior of the monodromy operator Nα of H̃(α)
when α tends to 0.
The asymptotic behaviors given by Theorem 5.3 are useful for the study of the asymptotic behaviors of the non-archimedean components of regulators and height pairings in
degeneration. This is because the non-archimedean components of the regulators and
height pairings are described by monodromy operators.
Remark 5.7. (1) Slightly refining the above proof of Theorem 5.3, we can show that
the function ν 7→ t(ν)−1 N(ν)t(ν) on x[:],1 extends to a complex analytic function in
ν(fj,λ , fj )1/2 and ν(fj+1 , fj )1/2 on an open neighborhood of µ in x[:] . We do not pursue
this here.
(2) This Section 5 could be presented as a story of the fs monoid (Mx /Ox× )x and an
admissible representation over C of its dual cone, and of its space of ratios, forgetting the
fs log scheme x and the prime number ℓ. But the authors prefer the above presentation,
for they hope to apply the result to ℓ-adic sheaves which come from motives over an
algebraic variety over a non-archimedean local field, by the method described in 5.6.
6
Higher direct images
6.1. For the Riemann hypothesis part of the Weil conjecture, the proofs given in [3] and
in [4] are to consider higher direct images of mixed sheaves and reduce the problem to the
study of higher direct images in the case of relative dimension one. We hope the study of
higher direct images for the category AX is important and that the monodromy-weight
conjecture is reduced to the case of relative dimension one.
Here we present an attempt in this direction.
6.2. Let X be an fs log scheme of finite type over a finite field.
Let AsX be the full subcategory of AX consisting of objects H such that grW
w H are
semi-simple for all m.
If the statement in the following Question 6.3 is true, our plan of the study in 6.1
would be a smooth route.
Question 6.3. Is the following statement true?
Let X → Y be a vertical projective log smooth saturated morphism of fs log schemes
of finite type over a finite field. Then for every i ≥ 0, the i-th direct image functor H i
sends AsX to AsY . Here we define Ww H i (H) as the image of H i (Ww−i H). It sends pure
objects of weight w to pure objects of weight w + m.
In the rest of this Section 6, we consider the case where Y is a standard log point and
X is a degenerate elliptic curve (6.4) hoping that it will be helpful for the future studies
of Question 6.3. Concerning this X → Y , we give Proposition 6.5 treating a case for
which the statement in Question 6.3 is true, and give a related result Proposition 6.11.
Logarithmic geometry and Frobenius
We give also an example of H in 6.17 which does not belong to AsX and whose H 1 does
not belong to AY .
6.4. We consider the following X → Y . Let K be a non-archimedean local field with
finite residue field k, let Y = Spec(k) with the standard log structure, and let X be the
special fiber of a projective model over OK with semi-stable reduction of a Tate elliptic
curve over K. Regard X as a log smooth vertical saturated fs log scheme over Y in the
canonical way.
Proposition 6.5. Let X → Y be as in 6.4. Let H be an object of AX such that for every
w ∈ Z, grW
w H is the pullback of an object of AY .
(1) The higher direct images of H belong to AY .
(2) If H belongs to AsX , the higher direct images belong to AsY .
We expect that the proof of Proposition 6.5 given below works to extend Proposition
6.5 to a general X → Y of relative dimension one. See Remark 6.10.
6.6. Let X → Y be as in 6.4. We denote Y also by y.
In 6.6 and 6.7, we give preparations on the log étale cohomology ([15]) and the log
fundamental group ([9] Section 10), respectively.
Let
1
T ∗ := Hlogét
(X ×Y y(log), Zℓ ),
which is a free Zℓ -module of rank 2 with an action of π1log (y) = Gal(y(log)/y). Let
T := Hom Zℓ (T ∗ , Zℓ ),
L := T ⊗Zℓ Qℓ ,
L∗ := T ∗ ⊗Zℓ Qℓ .
We denote the group law of T multiplicatively, but the group law of T ∗ , L, L∗ additively.
We regard L as the Lie algebra over Qℓ of the ℓ-adic Lie group T . It is a commutative
Lie algebra, and the inclusion map T → L is thought as the logarithm.
We have a Zℓ -base (γj )j=1,2 of T , a lifting F ∈ π1log (y) = Gal(y(log)/y) of the q-th
power map in Gal(k/k), and a topological generator γ0 of Gal(y(log)/y) such that the
action of π1log (y) on T satisfies
F (γ1 ) = γ1 ,
F (γ2 ) = γ2q ,
γ0 (γ1 ) = γ1 γ2 ,
γ0 (γ2 ) = γ2 ,
where q is the order of the finite field k. As representations of π1log (y), we have an
isomorphism L∗ ∼
= S1 , where S1 is as in 1.9, such that the base (ej )j of S1 in 1.9 is the
image of the dual base of (γj )j=1,2 in T ∗ by this isomorphism.
6.7. Let X → Y be as above.
We consider the logarithmic fundamental group π1log (X). Fixing a closed point x of X
and considering the stalk at x(log) lying over y(log), a smooth Qℓ -sheaf on the log étale
site Xlogét of X is identified with a representation of π1log (X) over Qℓ .
We have an exact sequence
6.7.1.
1 → π1log (X ×Y y(log)) → π1log (X) → π1log (y) → 1,
Logarithmic geometry and Frobenius
which splits (π1log (X) is the semi-direct product). We have a surjective homomorphism
π1log (X ×Y y(log)) → T whose kernel is a pro-ℓ′ group, where a pro-ℓ′ group means the
inverse limit of finite groups whose orders are coprime to ℓ. Thus we have a surjective
homomorphism from π1log (X) to the semi-direct product of T and π1log (y) whose kernel is
a pro-ℓ′ group.
6.8. We discuss how to compute the higher direct images. We use the Lie algebra cohomology. When we consider a unipotent representation of T , we consider the Lie action of
the Lie algebra L by taking the logarithm of the action of T .
Assume that the action of π1log (X ×Y y(log)) on (the stalk of) H is unipotent. Let
6.8.1.
0 → H → H ⊗Qℓ L∗ → H ⊗Qℓ ∧2 L∗ → 0
be the standard complex to compute the Lie algebra cohomology of L with coefficients
in H. Here we identify H with its stalk endowed with the action of π1log (X). The first H
is put in degree 0 of the complex. This 6.8.1 is a complex of representations of π1log (X)
(which acts on L∗ through π1log (X) → π1log (y)) and hence is a complex of ℓ-adic sheaves
on Xlogét .
As an ℓ-adic sheaf, the i-th higher direct image H i (H) of H on Y is identified, by
taking the pullback from Y to X, with the i-th cohomology of this complex.
6.9. We prove Proposition 6.5.
We identify L∗ with the object S1 of AY pure of W -weight 1 (6.6) and regard it as
an object of AX by pullback. Then each term of the complex 6.8.1 with the W of the
tensor product is regarded as an object of AX . Furthermore, by the assumption on grW H
in Proposition 6.5, morphisms in 6.8.1 are morphisms in AX . Thus 6.8.1 is a complex in
AX .
As an object of BY , the i-th higher direct images of H on Y is identified with the i-th
cohomology of this complex. Since AX is an abelian category, these higher direct images
are objects of AX . We prove that this object on Y belongs to AY . Take a closed point
x of X whose log structure is strict over Y . Since the local monodromy of the higher
direct image at x is unipotent, the local monodromy of the higher direct images on Y
is unipotent. Since the relative monodromy filtration of a non-trivial element of σ(x) on
the higher direct image (regarded as an object of AX ) is the Frobenius weight filtration
∼
=
and since σ(x) → σ(y), the relative monodromy filtration of a non-trivial element of σ(y)
on the higher direct image is the Frobenius weight filtration. This shows that the higher
direct images belong to AY .
We prove (2) of Proposition 6.5. We may assume that H is W -pure and hence H comes
from a semi-simple object of AY by pullback. Since L∗ ∼
= Sr+1 ⊕ Sr−1
= S1 and Sr ⊗ S1 ∼
i ∗
for r ≥ 0, Proposition 1.10 shows that H ⊗ ∧ L are semi-simple for all i. Since the i-th
higher direct image of H is a subquotient of H ⊗ ∧i L∗ , it is semi-simple.
Remark 6.10. It seems that the above proof of Proposition 6.5 works for the general
case of relative dimension one. We take L to be the Lie algebra of an ℓ-adic nilpotent
quotient of π1log (X ×Y y(log)) (then L need not be commutative in this general situation,
and L∗ becomes of W -weight ≥ 1, not necessarily of W -weight 1). However, for the proof
of this generalization, it seems that we have to discuss the AX -version of the theory of
Logarithmic geometry and Frobenius
mixed Hodge structures on nilpotent quotients of the fundamental groups of an algebraic
varieties in [5]. We hope to discuss this elsewhere.
An interesting point of the following proposition is that the projectivity of X → Y
which appears in Question 6.3 also appears in the condition (iii) in (3).
Proposition 6.11. Let X → Y be as in 6.4 and let H be a W -pure simple object of AX .
×
(1) There is c ∈ Qℓ such that on the stalk of H, the action of γ1 − c is nilpotent.
(2) If c in (1) is not 1, the higher direct images H i (H) vanish for all i.
(3) If c in (1) is 1, the following four conditions are equivalent.
(i) The higher direct images H i (H) belong to AY for all i.
(ii) H 0 (H) belongs to AY .
(iii) The Lefschetz class map H 0 (H) → H 2 (H(1)) is an isomorphism of ℓ-adic sheaves
on Y .
(iv) H is the pullback of an object of AY .
If these equivalent conditions are satisfied, (i) is satisfied with AY replaced by AsY and
the map in (iii) becomes an isomorphism in AY .
We give preparatory lemmas to prove Proposition 6.11.
Lemma 6.12. Let X → Y be as in 6.4 and let H be an object of AX . Then the action
of γ2 on the stalk of H is unipotent.
Proof. Let α be a singular point of X. We show that γ2 comes from the local monodromy
group at α. Because the local monodromy of H is unipotent, this will show that the
action of γ2 is unipotent. We prove that γ2 is in the image of π1log (α ×y y(log)) → T . This
1
map is dual to T ∗ → Hlogét
(α ×y y(log), Zℓ ) ∼
= Zℓ , which sends e2 to 1 and e1 to 0. Hence
the last map is induced by γ2 .
Lemma 6.13. Let X → Y be as in 6.4
Land let H be an object of AX . Then we have a
unique direct sum decomposition H = c I(c) in A, where c ranges over all elements of
×
Qℓ , such that on the stalk of I(c), the action of γ1 − c is nilpotent.
Proof. Let Vc be the generalized eigenspace of γ1 for the eigenvalue c in the stalk of H.
We prove that Vc is preserved by the action of π1log (X). The Frobenius F and γ2 commute
with γ1 , so they preserve Vc . We prove γ0 preserves Vc . From γ0 γ1 γ0−1 = γ1 γ2 , we have
(γ1 − c)n γ0−1 = γ0−1 ((γ1 − c)γ2 + c(γ2 − 1))n for n ≥ 0. If n is sufficiently large, since the
action of γ2 − 1 is nilpotent (Lemma 6.12), ((γ1 − c)γ2 + c(γ2 − 1))n kills Vc and hence
(γ1 − c)n kills γ0−1 Vc . Hence γ0−1 preserves Vc .
The projections to the generalized eigenspaces are given by a polynomial of γ1 and
hence preserve relative monodromy filtrations, and hence they give a direct decomposition
in AX .
Lemma 6.14. Let X → Y be as in 6.4, let H be an object of AX , and assume that
H = I(c) (Lemma 6.13) with c 6= 1. Then H i (H) = 0 for all i.
Proof. As a Qℓ -vector space, the stalk of H i (H) is identified with the i-th cohomology of
the complex 0 → H → H ⊕ H → H → 0, where H denotes the stalk of H, H → H ⊕ H is
(γ1 − 1, γ2 − 1), and H ⊕ H → H is (1 − γ2 , γ1 − 1). It is acyclic because γ1 − 1 : H → H
is an isomorphism.
Logarithmic geometry and Frobenius
6.15. We prove Proposition 6.11.
(1) follows from Lemma 6.13, and (2) follows from Lemma 6.14.
We prove (3).
The implication (i) ⇒ (ii) and its version for AsY are clear.
We consider the actions of Nj = log(γj ) (j = 1, 2) on the stalk of H. Note that N1 is
of Frobenius weight 0 and N2 is of Frobenius weight −2.
We prove the implication (ii) ⇒ (iv). We have H 0 (H) 6= 0 because there is a non-zero
element x of the lowest Frobenius weight part of the stalk of H such that N1 (x) = 0
(N2 (x) = 0 automatically). Since H 0 (H) ∈ AY , we have H 0 (H) ∈ AX . Since H is
simple, we have H = H 0 (H), that is, (iv) is satisfied.
We prove the implication (iii) ⇒ (iv). As Qℓ -vector spaces, we can identify H 0 (H) with
the part H N1 =N2 =0 of H (this H denotes the stalk of H), H 2 (H) with H/(N1 H + N2 H),
and the Lefschetz class map with the map induced by the identity map of H. Then the
assumption that it is an isomorphism implies N1 = N2 = 0. In fact, if Nj 6= 0 for some j,
H Nj =0 ∩ Nj H is a non-zero subspace, on which N3−j acts. Hence H N1 =N2 =0 ∩ Nj H 6= {0}.
Hence H comes from Y . This object on Y belongs to AY because its pullback H belongs
to AX .
The implication (iv) ⇒ (i) follows from Proposition 6.5.
We prove the implication (iv) ⇒ (iii). This is reduced to the case where H is the
constant object Qℓ . In this case, H 0 (H) = Qℓ of pure W -weight 0, H 2(H) = Qℓ (−1) of
pure W -weight 2, and the Lefschetz map is identified with the identity map Qℓ → Qℓ .
6.16. The authors expect that the equivalent conditions in Proposition 6.11 (3) are actually always satisfied. If this is the case, the statement in Question 6.3 is true for X → Y
in 6.4. To see this, by Lemma 6.13, we may assume H = I(c) for some c. If c 6= 1, it
follows from Lemma 6.14. If c = 1, using (iv) in Proposition 6.11 (3), it follows from
Proposition 6.5.
6.17. An example for non-semi-simple case which does not have the property in the
statement in Question 6.3.
Let X → Y be as in 6.4. Let H be the two dimensional Qℓ -space with base (fj )j
endowed with the following action of π1log (X) over Qℓ . Take a splitting of the exact
sequence 6.7.1, let the action of π1log (y) be the trivial one, and let the action of π1log (X ×Y
y(log)) be the one induced from the action of T , where γ2 acts trivially and γ1 acts as
γ1 (f1 ) = f1 , γ1 (f2 ) = f2 + f1 . Let W of H be pure of weight 0.
This H is a W -pure object of AX but is not semi-simple (f1 generates a subobject in
AX which is not a direct summand). This H does not satisfy the statement in Question
6.3. The first higher direct image H 1 (H) of H does not belong to AY . This is seen as
follows.
Note that H 1 (H) has the pure W -weight 1 by the definition of the W of the higher
direct image. We identify L∗ with S1 (6.6). The map H → H ⊗ L∗ in 6.8.1 sends f2 to
f1 ⊗ e1 and sends f1 to 0. From this, we see that H 1 (H) is identified with the subquotient
P/Q of H ⊗ L∗ , where P has the base f2 ⊗ e1 , f1 ⊗ e2 , f1 ⊗ e1 and Q has the base f1 ⊗ e1 .
Hence H 1 (H) has the base f2 ⊗ e1 mod Q, f1 ⊗ e2 mod Q, the former has Frobenius
weight 0 and the latter has Frobenius weight 2. Since γ0 (fj ) = fj , γ0 (e1 ) = e1 , and
γ0 (e2 ) ≡ e2 mod Qℓ e1 , we have γ0 (f2 ⊗ e1 ) = f2 ⊗ e1 and γ0 (f1 ⊗ e2 ) ≡ f1 ⊗ e2 mod Q, and
Logarithmic geometry and Frobenius
hence γ0 acts trivially on H 1 (H). Hence the relative monodromy filtration of a non-zero
element of σ(y) (which acts as zero) on H 1 (H) is pure of weight 1, and does not coincide
with the Frobenius weight filtration. Hence H 1 (H) is not an object of AY .
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Logarithmic geometry and Frobenius
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Kazuya KATO
Department of mathematics
University of Chicago
Chicago, Illinois, 60637, USA
kkato@uchicago.edu
Chikara NAKAYAMA
Department of Economics
Hitotsubashi University
2-1 Naka, Kunitachi, Tokyo 186-8601, Japan
c.nakayama@r.hit-u.ac.jp
Sampei USUI
Graduate School of Science
Osaka University
Toyonaka, Osaka, 560-0043, Japan
usui@math.sci.osaka-u.ac.jp